Rejection Region Approach to Hypothesis Testing for
One Proportion Problem

One can perform hypothesis testing using the p-value approach, or one can
perform hypothesis testing using a rejection region approach. The
conclusions from the two approaches are exactly the same.

There are six parts of a test when using the rejection region approach:

Null and alternative hypotheses

Level of significance

Test statistics

Critical values and rejection region

Process of checking to see whether the test statistic falls in the rejection
region

Conclusion in words

Test statistic: The sample statistic one uses to either reject Ho (and conclude Ha) or not to reject Ho.

Critical values: The values of the test statistic that separate
the rejection and non-rejection regions.

Rejection region: the set of values for the test statistic that
leads to rejection of Ho.

Non-rejection region: the set of values not in the rejection region
that leads to non-rejection of Ho.

As mentioned in lesson 8, the logic of hypothesis testing is to reject
the null hypothesis if the sample data are not consistent with the null
hypothesis. Thus, one rejects the null hypothesis if the observed test statistic
is more extreme in the direction of the alternative hypothesis than one
can tolerate. The critical values are the boundary values obtained corresponding
to the preset level.

One-proportion Z-test for

Step 0. Check the conditions for the one-proportion z-test to be valid:

no 5

n(1 - o) 5

Step 1. Set up the hypotheses as one of:

Two-tailed

Right-tailed

Left-tailed

Ho: = o

OR

Ho: = o

OR

Ho: = o

Ha: o

Ha: > o

Ha: < o

Step 2. Decide on the significance level, .

Step 3. Compute the value of the test statistic:

Step 4. Find the appropriate critical values for the tests using the z-table.
Write down clearly the rejection region for the problem.

Step 5. Check to see if the value of the test statistic falls in the rejection
region. If it does, then reject Ho (and conclude Ha).
If it does not fall in the rejection region, do not reject Ho.

Step 6. State the conclusion in words.

Some expert claims that the probability of each person being left-handed
is 0.25. It is observed that out of 30 randomly sampled people, 10 are left-handed.
Using = 0.05, is there sufficient evidence to conclude that the population proportion
is different from 0.25?

a. Use the rejection region approach to perform the testing.

Step 0. Can we use the one-proportion z-test?

The answer is yes since the hypothesized value o is 0.25 and we can check that:

no = 30 · 0.25 = 7.5 5,n(1 - o)
= 30 · (1 - 0.25) = 22.5 5.

Step 1. Set up the hypotheses (since the research hypothesis is to
check whether the proportion is different from 0.25, we set it up as
a two-tailed test):

Ho: = 0.25Ha: 0.25

Step 2. Decide on the significance level, .

According to the question, = 0.05.

Step 3. Compute the value of the test statistic:

Step 4. Find the appropriate critical values for the test using the
z-table. Write down clearly the rejection region for the problem. We
can use Table 2 to find the value of Z0.025 since
the row for df = (infinite) refers to the z-value.

From Table 2, Z0.025 is found to be 1.96 and thus
the critical values are ± 1.96. The rejection region for the two-tailed
test is given by:

z > 1.96 or z < -1.96

Step 5. Check whether the value of the test statistic falls in the
rejection region. If it does, then reject Ho (and
conclude Ha). If it does not fall in the rejection
region, do not reject Ho.

The observed z-value is 1.05 and will be denoted as z*.
Since z* does not fall within the rejection region, we do not
reject Ho.

Step 6. State the conclusion in words.

Based on the observed data, there is not enough evidence to conclude
that the population proportion of left-handed people is different
from 0.25.

b. Use the p-value approach to perform the testing.

Step 0 - Step 3. The first few steps (Step 0 - Step 3) are exactly
the same as the rejection region approach.

Step 4. In Step 4, we need to compute the p-value. Since it
is a two-tailed test: